Question

Let the conditions of the 360 theorem hold such that ray x, ray y and ray z all have the same end point, none of them are collinear, and that no rays lie in the interior of the other two. The crucial step in the proof of the 360 theorem was that among the opposite rays of x, y and z at least one of these opposite rays goes through the interior formed by the other two rays. Prove that all of these opposite rays lie in the interior formed by the other two rays.(Axiom geometry, prove with 360 theorem and prove without 360 theorem)

Answer #1

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

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